Basic skills --- fast arithmetic, fast algebra, fast geometry, and fast word problem solving.

               Advanced strategies --- reverse operations, backward reasoning, pattern recognition, Achilles' Heel tactic, and multi-dimension attacks.

          We teach students critical thinking skills in our programs based on our many years of research and university teaching experiences. We illustrate some of our approaches in the word problem solving context as shown in our videos. The same principles and methods can be applied to other areas of standardized math tests.

          Since the number of different problems that can appear in those tests is potentially infinite, we cannot just teach students how to solve each different problem. Instead, we have to teach students the skills to solve the problems. We know that the College Board recycles entire SAT tests that have already been given to your older brothers and sisters. Of cause we know how to use that fact to raise your scores, but that does not change our critical thinking approaches in our training programs. 

          Give people fish, you have fed them for today; Teach people to fish, you feed them for a lifetime. Therefore, in order to have a perfect score, SAT Math 800, for an example, one has to learn the critical thinking skills. In the word problem solving context, it is the skill to solve a new different word problem using the knowledge of old problem when the two problems are involving the same mathematical equation. 

          Despite of many years schooling, many students are not taught the critical thinking skills in our school systems. In the words of Kelly Price, a curriculum coordinator in Forsyth School District, Georgia, "Some students were good at the other way of doing math because all they had to do was memorize and regurgitate," she said. "They never applied or understood, but they were good at spitting it back out" (Dodd 2010).

          Willingham (2007) points out the nature of the problem: "Typically, the students are focusing on the scenario that the word problem describes (its surface structure) instead of on the mathematics required to solve it (its deep structure). So even though students have been taught how to solve a particular type of word problem, when the teacher or textbook changes the scenario, students still struggle to apply the solution because they don't recognize that the problems are mathematically the same."

          "Can critical thinking actually be taught? Decades of cognitive research point to a disappointing answer: not really" (Willingham 2007). Unfortunately, this is supported by the test results. Fewer than 25% of 2010 graduates who took the ACT are college-ready. "Despite a decade of high-school reform efforts, including breaking larger high schools into smaller ones and pushing more students into college-prep courses, there is still no solid evidence on how best to boost achievement" (Banchero 2010).

          We are not surprised by all these given our experiences teaching college students. After all, on the state tests in New York, students can get half-credit if they answer 15 cents instead of 18 cents by subtracting 57 cents from three quarters (Campanile and Edelman 2010). Many states, however, have been doing a "smarter" way to boost both self-esteem and proficiency rates under No Child Left Behind law: dumbing down the state tests (Cronin, Dahlin, Adkins, and Kingsbury 2007) to give a false impression of education success.

          The success of our test prep programs is measured by the results of the SAT, ACT, GMAT, and GRE math tests taken by our students. Because the standards of those tests are stable year over year, they tell us how well we teach critical thinking skills. By inventing and using our exclusive word problem solving apparatus, we are able to challenge the above mentioned Willingham’s (2007) notion that critical thinking skills cannot be taught. This is because we teach students to connect the surface structure (distance of travelling, the speed, and time travelled) with the deep structure (algebra equation) by touching the objects, manipulating the variables, and seeing the movements in action. When Students work on another surface structure (amount of work done, the rate that work is done, and time worked), they make such connections again.

          To help students make the connections, our apparatus links the surface structure with algebraic variables and equations and is applicable to seven common types of word problems that frequently appear on the SAT Math test -- distance, mixtures, coins, age, levers, finance, and work problems.

          By using our invention, students make the connections repeatedly in different surface structure word problems, which all point out to the same deep structure. They will eventually look beyond the surface structures and focus on the two kinds of relations: the first is the relation between the words describing the variables, such as the speed vs. R and time vs. T. The second is mathematical relations between the variables.

          Students thus develop the skills to solve the new different problems with the knowledge of the old problems, first by using our invention, then without the tool, all in a quick and fun way to solve word problems. We look forward to making critical thinking and problem solving a passion in your life no matter what career path you take. You can start by working on our sample math problems and contacting us if you have any questions.     

References:

Banchero, Stephanie (2010), "Scores Stagnant at High Schools", the Wall Street Journal , Education, August 18, 2010.

Campanile, Carl and Susan Edelman (2010), "NY passes students who get wrong answers on tests", New York Post, June 6, 2010.

Cronin, John, Michael Dahlin, Deborah Adkins, and G. Gage Kingsbury, (2007), "The Proficiency Illusion", Thomas B. Fordham Institute, Report, October 2007.

Dodd, D. Aileen (2010), “Final exams show teens struggle with new math curriculum”, the Atlanta Journal-Constitution, July 29, 2010.

Willingham, Daniel (2007), "Critical Thinking Why Is It So Hard to Teach?" American Educator, Summer 2007 pp. 8-18.